Fourier rokuchinja. Fast Fourier rokuchinja. Discrete Fourier rokuchinja

Fourier Kushandurwa - kusanduka, nokuwadzana imwe kushanda ari shanduka chaiyo. oparesheni iyi rinoitwa nguva dzose tinoona ruzha siyana. Nzeve unobereka otomatiki "kukarukureta", iro basa redu kuziva anogona chete pashure Kuongorora Chikamu dzakakwirira masvomhu. kunzwa Nhengo ari kusanduka yevanhu constructs, umo ruzha (kwakajairika vibrational inofamba pakati inoramba iri kwakavharwa nenyama unosvikirwa, izvo hunopararira pakati chinozunguzirwa chimiro yakasimba, yakaita mvura mvura kana gaseous svikiro) anopiwa mune zvakawanda akatevedzana tsika ruzha pamwero zvenzwi kunosiyana dzakakwirira. Shure kwaizvozvo, uropi anochinja mashoko kunyika yose yaizivikanwa inzwi.

Kwemasvomhu Fourier rokuchinja

Kutendeutsa ruzha kana zvimwe vibration pezvakaitwa (nechiedza achine uye gungwa nenyika uye kuti akatisiyira kana rezuva zvinogara) kunogona kuitwa uye pachishandiswa kwemasvomhu nzira. Saka, achishandisa nzira iyi, iyo mabasa anogona kuwedzerwa nokusuma vibrational pezvakaitwa wakaiswa pakati sinusoidal zvinoriumba, i.e. Wavy makakombama izvo enda kubva zvishoma kuti vaifanira uye zvakare kushoma, kufanana nefungu regungwa. Fourier kushandurwa - kuchinja basa iro rinotsanangura danho kana amplitude mumwe sinusoid chaienzanirana imwe maitikiro. Danho iri pokutangira pamusoro munyonga, uye amplitude - vayo yakakwirira.

Fourier rokuchinja (mienzaniso vanoonekwa vari Photo) riri simba zvikuru, izvo rinoshandiswa zvesayenzi. Mune zvimwe zviitiko, rinoshandiswa sechiratidzo yokugadzirisa equations asi kunzwisisa inotsanagura simba pezvakaitwa kuitika pasi pokudzora chiedza, kupisa kana zvemagetsi simba. Mune zvimwe zviitiko, zvinoita kuti kutsanangura dzose zvinoriumba mu waveforms yakaoma, nokuda ichi kunogona kuva zvechokwadi kududzira siyana experimental achicherechedza kemisitiri, mushonga uye zvenyeredzi.

nhoroondo mashoko

Munhu wokutanga kushandisa nzira iyi yaiva yemasvomhu French Zhan Batist Fure. Kutendeuka, vakabva zita pashure pake, pakutanga kushandiswa kutsanangura kupisa conduction mashandiro. Fourier akura upenyu hwake hwose tichiita kudzidza ehupfumi kupisa. Akaita mupiro kukuru kune kwemasvomhu dzidziso kutsunga nemidzi algebraic equations. Fourier aiva purofesa kuongorora panguva Ecole Polytechnique, Munyori Institute of Egyptology, aiva paakaenda basa, izvo zvakakonzera mhirizhonga panguva kuvakwa inoenda nokuTurin (pasi utungamiriri wake vakapedzwa vanopfuura zviuru 80 mativi makiromita ane malarial anezvimiti). Zvisinei, kwekuratidzira zvose izvi hazvina kumisa musayendisiti vaiita kwemasvomhu kuongorora. Muna 1802 zvakanzi hwokurara ive mukana wokuti vanhu vanga kuti rinotsanangura Mashoko kupisa mune zvinhu zvakaoma. Muna 1807, musayendisiti akawana nzira nokuda nekugadzirisa mukana wokuti vanhu vanga ichi, izvo pakuzivikanwa se "Fourier rokuchinja".

chafariz conductivity ongororo

Vatsvakurudzi akashandisa kwemasvomhu nzira kutsanangura kupisa conduction mashandiro. A muenzaniso rakafanira, sezvamakaropafadzwa pasina kunetseka computation ndiyo Mashoko chafariz simba nemunhu mhete simbi, chikamu chimwe achinyatsotaurira moto. Kuti aite kuedza Fourier Hot chikamu mhete uye akamuviga mujecha yakaisvonaka. Zvadaro, tembiricha kuyerwa vakatakura pamusoro pakatarisana chacho. Pakutanga, kupisa kuparadzirwa iri pamutemo: chikamu mhete - chando, uye mumwe - inopisa, pakati zvikamu zvinogona vachengete rakapinza tembiricha gradient. Zvisinei, panguva kupisa kuparadzirwa mhiri simbi pamusoro, zvinoramba zvichitowedzera kukosha yunifomu. Saka, nenguva isipi, izvi zvinotora chimiro sine wave wave. First girafu zvishoma kunowedzera uye zvinoderedza zvakanaka, zvakarurama nemitemo kusiyana pamusoro cosine kana sine wave kushanda. Wave zvishoma equalized uye somugumisiro tembiricha anova yunifomu pamusoro pepasi rin'i.

Muvambi nzira iyi vakafunga kuti kwokutanga kuparadzirwa rakanyatsojeka kwemashifiti inogona yaora mumitauro chepuraimari sine wave nemafungu. Mumwe nomumwe wavo achava danho rayo (pakutanga nzvimbo) uye ayo tembiricha kunonyanyisa. Saka mumwe nomumwe kwakadaro zvinoriumba kuchinja kubva zvishoma kuti vaifanira uye shure kupedza chimurenga kumativi rin'i integer nguva. Chinoumba kuva nguva yainzi chinokosha harmonic, uye mutengo maviri kana kupfuura nguva - wechipiri uye zvichingodaro. Somuenzaniso, imwe kwemasvomhu basa kuti kurondedzera kunonyanyisa tembiricha, Pasika kana nzvimbo inonzi Fourier rokuchinja pamusoro kugovera basa. Scientist akauya chete chinoumba kuti kwakaoma sokurondedzera akarongeka, nokuti nyore kushandisa maturusi - mitsara Sine uye cosine, mune mari yekupa kutanga kuparadzira.

Musimboti ari kuongorora

Kushandisa kuongorora uku kutendeuka kupisa pakugoverwa pamusoro yakasimba chinhu, ane annular chimiro, mudzidzi wemasvomhu akafunga kuti kuwedzera nguva sinusoidal zvinoriumba kukonzera ayo nokukurumidza damping. Izvi zvinonyatsooneka pane huru uye wechipiri harmonics. Kwokupedzisira tembiricha kunosvika kaviri kufadza uye shoma tsika mune rimwe vanopfuura, uye pakutanga - chete kamwe. Zvakaratidza kuti daro vaifamba nokupisa yechipiri harmonic iri hafu yacho core. Uyewo, gradient rechipiri hafu achava steeper pane kwekutanga. Naizvozvo, sezvo zvikuru chafariz nekubudisa rinopfuura chirikadzi usingashandi kure, ipapo ichi chichange damped harmonic nguva zvina dzichimhanya kupfuura huru, ave kushanda nguva. In zvinotevera izvi zvichava kunyange nokukurumidza. Yemasvomhu aidavira kuti nzira iyi kunoita kuti ngaaverenge muitiro kwokutanga kuparadzirwa tembiricha pamwe nguva.

Call vakararama

Fourier rokuchinja algorithm rava dambudziko nheyo yokungotaura wemasvomhu panguva iyoyo. Mukutanga namapfumbamwe remakore, vakawanda masayendisiti vakakurumbira, kusanganisira Lagrange, Laplace, Poisson, Legendre uye Biot haana kubvuma zvaaitaura kuti tembiricha iri pakutanga nokuparadzirwa iri yaora kupinda zvinoumba muchimiro chinokosha chinozunguzirwa uye yakakwirira maitikiro. Zvisinei, iye Academy of Sciences vaisagona kukanganwa migumisiro vakawana yemasvomhu, uye paakavapa iye Mubayiro nekuda dzidziso kupisa conduction mitemo, uyewo vachiitisa kuenzanisa wayo chaiwo kuedza. Mune Fourier nzira, chikuru ari kuramba ndechokuti munhu discontinuous basa inomiririrwa chitsama anoverengeka sinusoidal mashandiro, zviri kuramba. Pashure pezvose, ivo rinorondedzera anobuda twasu uye curved nemitsetse. musayendisiti Contemporary kumbobvira vasangana ezvinhu akadaro, apo discontinuous mabasa yakarondedzerwa mubatanidzwa kwechiropa, akadai quadratic, Linear, sine wave kana exhibitor. Kuri kuti yemasvomhu zvakanaka ake chaitika, Verengai isingaperi munhevedzano trigonometric mashandiro anofanira kungogumira chaiyo nokukurumidza. Kunyange zvazvo kutaura sokuti penga. Zvisinei, pasinei kusava nechokwadi nevamwe vatsvakurudzi (semuenzaniso Claude Navier, Sofi Zhermen) vaikudza kukura kwebasa kutsvakurudza uye akavabudisa kunze Kuongororwa nokuparadzirwa kupisa. A masvomhu, Zvichakadai, vakaramba kutambura mubvunzo kana chitsama anoverengeka sinusoidal mashandiro ipfupi kuti mufananidzo chaiwo anobuda.

Nhoroondo-gore 200

Dzidziso iyi yakatanga pamusoro maviri emakore, nhasi zviri pakupedzisira kuumbwa. Tichibatsirwa ari spatial kana epanyama mabasa apwanyika kupinda sinusoidal zvinoriumba kuti une kakawanda, danho uye amplitude. kutendeuka uku kunowanikwa vaviri kwemasvomhu dzakasiyana nzira. Wokutanga wavo rinoshandiswa zvakadyiwa apo anokonzera iri kuramba basa, uye wechipiri - panyaya ndipo inomiririrwa muzvizhinji kuti discrete nomumwe kuchinja. Kana mashoko inotengwa kubva tsika, izvo anotsanangurwa pana discrete zvikamu, rinogona rakakamurwa akawanda discrete sinusoidal frequencies mashoko - kubva apasi ipapo kaviri, katatu, uye nezvimwewo pamusoro chinokosha. mari iyi inonzi iri Fourier akatevedzana. Kana kutanga mashoko anogadza kukosha neimwe nhamba chaiyo, rinogona dzichakoromorwa kupinda vakawanda sinusoidal zvinogoneka frequencies zvose. Inonzi Fourier chinokosha, uye sarudzo kunoreva Kushandurwa chinokosha basa. Pasinei nzira chokuwana kusanduka, mumwe nomumwe kakawanda anofanira kuratidza nhamba mbiri: amplitude uye maitikiro. tsika aya anoratidzwa sezvo rimwe kunzwisisa nhamba. Mashoko akaoma variables dzidziso pamwe Fourier kuchinja kuita Masvomhu akabvumira magadzirirwe siyana yemagetsi mumatunhu, ari kuongorora zvemaziso vibrations, kudzidza chinozunguzirwa kupararira mashandiro uye mumwe.

Fourier rokuchinja nhasi

Mazuva ano, kudzidza izvi rinonyanya mamota pasi nokuwana nzira dzinoshanda kuti kuchinja kubva basa kuichinja kudzoka pfungwa. Izvi mhinduro inonzi zvakananga uye Ivan Kuchin Fourier rokuchinja. Zvinorevei? Kuti kuona chinokosha uye kuti zvakananga Fourier rokuchinja, unogona kushandisa kwemasvomhu nzira, asi unogona analytic. Pasinei chokwadi kuti kana vari kushandiswa mune tsika pane zvinetso, integrals vakawanda akawanikwa akapinda kwemasvomhu handbooks kare. Achibatsirwa kusanganiswa nzira inogona dzaona mashoko, chimiro iyo inobva experimental data, mumwe basa ane integrals ari pamabwendefa zvisipo, uye vane zvakaoma kufungidzira munzvimbo Analytical muchimiro.

Kusati kwatombova nekombiyuta okugadzira Masvomhu transformations yakadaro richikunetesa kwazvo, ivo zvinoda Chinyorwa vakauraya vazhinji svomhu akavhiyiwa kuti kuvimba uwandu pfungwa zvinorondedzera chinozunguzirwa ishande. Kuti zviite kuti nzvimbo nhasi, pane panorongwa, akabvumira kuti pave itsva nzira Analytical. Saka, muna 1965, Dzheyms Kuli uye Dzhon Tyuki akasika Software kuti akazozivikanwa "Fast Fourier rokuchinja". Kuponesawo nguva kukarukureta nokuderedza uwandu multiplications vari kuongorora zviri munyonga. "Fast Fourier rokuchinja" Nzira kunobva kugovera munyonga kupinda vakawanda dzakafanana mharidzo tsika. Saizvozvowo, nhamba multiplications ipfupi nehafu panguva imwe kuderedza pfungwa.

Kushandisa Fourier rokuchinja

Izvi inoshandiswa muminda siyana: In nhamba dzidziso, wefizikisi, chiratidzo kubudiswa, combinatorics, mukana dzidziso, cryptography, nhamba, oceanography, optics, acoustics, uye dzimwe geometries. Rich zvingangoitika kushandiswa kwayo dzinobva dzinoverengeka anobatsira features, iyo vanonzi "ehupfumi of Fourier kuchinja." Ngationgororei navo.

1. Kutendeuka basa iri Linear opareta uye runoenzanirana normalization iri unitary. pfuma iyi inonzi Parseval theorem no, kana ari mukuru zvazvingava, theorem no Plansherelja kana Pontrjagin hunyambiri.

2. kutendeuka iri reversible. Uyezve, zvakasiyana mugumisiro uri substantially chakafanana chimiro sezvo iri zvakananga nezvayo.

3. sinusoidal zvinhu mashoko avo pachavo kusiyaniswa mabasa. Izvi zvinoreva kuti kumiririrwa kwakadaro Kuchinja Linear equations vane kugara coefficients mune kwakajairika algebraic.

4. Maererano "convolution" theorem no, muitiro anoita kunzwisisa kuvhiyiwa chepuraimari chokuwanziridza.

5. Discrete Fourier rokuchinja anogona kukurumidza yakagadzirwa pakombiyuta uchishandisa "kutsanya" nzira.

Misiyano iri Fourier rokuchinja

1. Most kazhinji izwi rinoshandiswa kureva kuramba kusanduka, ichipa chero quadratically integrable kutaura sezvo Verengai kunzwisisa exponential rokuti zvimwe chisina kutsetseka frequencies uye amplitudes. Miti iyi ine dzakasiyana siyana, izvo zvingave zvakasiyana kuramba coefficients. The kuramba Nzira rinosanganisira tafura kutendeuka, iyo inogona kuwanikwa kwemasvomhu handbooks. A generalized nyaya ndiyo fractional kutendeuka, apo izvi zvinogona kumutswa kune vaida simba chairo.

2. kuramba Nzira iri zvavo vekare unyanzvi pakati Fourier dzakatevedzana kurondedzerwa upi Periodic kushanda kana mashoko, izvo zviri zvishoma nzvimbo uye vanomiririra pavo dzakatevedzana sinusoids.

3. Discrete Fourier rokuchinja. Nzira iyi rinoshandiswa kadambari kuti computation chesayenzi uye digitaalinen chiratidzo kugadzira. Kuti tiite izvi mhando kukarukureta kunodiwa kuti mashandiro dzokuziva pamusoro discrete akagadzirwa munhu pfungwa, nguva nenguva kana dzisina munharaunda pane kuramba Fourier integrals. Chiratidzo kutendeuka pane iyi anomiririrwa sezvo chitsama sinusoids. Kushandiswa "kutsanya" nzira rinobvumira kushandiswa digitaalinen negadziriso zvinangwa zvose zvinobatsira.

4. hwindo Fourier rokuchinja chinhu generalized maonero chaiwo nzira. Kusiyana mureza zvingaitwa kana chiratidzo chakarondedzera rinoshandiswa riri kubatwa yakazara-siyana nezvokuvapo shanduka uyu kufarira pano ndiyo chete kuparadzira kakawanda vemo apo tiine shanduka yepakutanga (nguva).

5. Vaviri-dimensional Fourier rokuchinja. nzira iri kushandiswa kushanda maviri-dimensional arrays kuti mashoko. Pakadaro, kutendeuka rinoitwa muna rimwe divi, uye ipapo - mune dzimwe.

mhedziso

Nhasi, ari Fourier nzira iri zvakasimba midzi mune zvesayenzi. Somuenzaniso, muna 1962 chakazarura Maumbirwe DNA kaviri akamonyoroka uchishandisa Fourier ongororo pachiimbwa X-ray diffraction. Recent crystals pfungwa DNA fibers, zvichiguma chifananidzo riri kuwanikwa diffraction, zvakanyorwa pamusoro firimu. Mufananidzo uyu akapa mashoko nezvokukosha kukosha amplitude nokushandisa Fourier rokuchinja kuti nekristaro ichi chivako. Danho Data vakawana nokuenzanisa DNA diffraction makadhi makadhi kuti vakawana muna Kuongororwa kwemakemikari zvivako zvakafanana. Somugumisiro, nyanzvi dzezvinhu zvipenyu akadzorera sekiristaro mamiriro - pakutanga basa.

Fourier rokuchinja kutamba chihombe basa kudzidza muchadenga, ivo wefizikisi ose semiconductor zvinhu uye plasma, microwave acoustics, oceanography, radar, seismology uye zvokurapa bvunzo.